A mortality model for a population In 

 which harvested Individuals do not 

 necessarily die: The stone crab 



Victor R. Restrepo 



University of Miami, Rosenstiel School of Marine and Atmospheric Science 

 Cooperative Institute for Marine and Atmospheric Studies 

 4600 Rickenbacker Causeway, Miami, Florida 33149 



Stone crabs Menippe mercenaria 

 support a valuable commercial fish- 

 ery in the Gulf of Mexico, with most 

 of the catch occurring near south- 

 west Florida. Florida landings in- 

 creased from about 400,000 lbs per 

 fishing season (15 October- 15 May) 

 in the early 1960s to an average 

 2.7 million lbs since 1988. The 1990 

 landings were valued at over $15 

 million. 



The stone crab fishery is unique 

 in that only the crabs' claws can 

 be harvested, provided that the 

 claws are of legal size (70 mm in 

 propodus length); declawed crabs 

 must be returned to the ocean. 

 Stone crabs can regenerate their 

 massive claws which contain much 

 of the crabs' edible meat (a large 

 claw can weigh 250 g). In a sense, 

 stone crabs are a "reusable re- 

 source" because claw regeneration 

 by previously declawed crabs ac- 

 counts for 1-10% of the annual 

 landings (Savage et al. 1975, Ehr- 

 hardt and Restrepo 1989). 



The main difficulty associated 

 with estimating mortality rates in 

 this unique fishery is that existing 

 models are not applicable to the 

 crabs' population dynamics. Tradi- 

 tional fisheries models are usually 

 based on the equation (see Beverton 

 and Holt 1957), 



dN 

 dt 



(F + M)Nt, 



where N is the population size, t is 

 time, and F and M are the instan- 



taneous fishing and natural mortal- 

 ity rates. An implication of this 

 model is that all harvested animals 

 die. Up to 50% of harvested stone 

 crabs may survive, depending on 

 fishing practices such as the amount 

 of time animals are exposed to air 

 and on the extent of the injury 

 caused by declawing (Davis et al. 

 1979). Therefore, the above model 

 is not appropriate for this fishery or 

 others like it. In this paper I develop 

 a mortality model that accounts for 

 the possibility that harvested indi- 

 viduals may survive. The model can 

 be used to estimate fishing mortal- 

 ity rates for stone crabs. 



The model 



Consider a closed population of large- 

 sized individuals (large enough to 

 lose both claws to fishing upon cap- 

 ture), in which catches are moni- 

 tored for a short period of time. 

 This time-period should be suffi- 

 ciently short to ensure that de- 

 clawed crabs will not have time to 

 regenerate their claws. Claw regen- 

 eration in large stone crabs takes 

 one year or more (Restrepo 1990), 

 so this should not be a major con- 

 straint. The population dynamics 

 during this time-period can be 

 modeled by subdividing the pop- 

 ulation into harvestable and unhar- 

 vestable crabs (those with and with- 

 out legal-sized claws, respectively). 

 Harvestable crabs may become un- 

 harvestable if they survive fishing; 

 unharvestable crabs may not be- 



come harvestable within the time- 

 interval, since it is assumed that 

 claw regeneration does not occur. 



Let 



^'N, "N = population sizes (in 

 numbers) of harvest- 

 able and unharvestable 

 crabs, 

 F = rate of capture (assum- 

 ed to be the same for 

 both types of crabs), 



^M, "M = natural mortality rates 

 for harvestable and un- 

 harvestable crabs, and, 

 S = fraction of harvestable 

 crabs that survive claw 

 removal and release (0 

 <S<1). 



For simplicity, assume that har- 

 vest and natural mortality rates re- 

 main fixed during the time-period. 

 Note also that unharvestable crabs 

 are immediately returned to the 

 water upon capture so that their 

 mortality due to capture is negli- 

 gible. The differential equations 

 describing the two-compartment 

 model are 



dt 



- (hM-HFS-HF(l-S))hNt 



= - (hM-(-F)hN,, and (la) 



d"Nt 



= - "M"Nt-HFShNt. (lb) 



Equation (la) is the standard mor- 

 tality model and simply shows that 

 crabs disappear from the population 

 due to fishing and natural mortali- 

 ty. Losses due to fishing are F'^Nt. 

 Of these, a fraction (1 - S) actually 

 die, and a fraction (S) become part 

 of the unharvestable population 

 (Eq. lb). Thus, F is a true fishing 

 mortality only when S = 0. 



Equation (la) has the general 

 solution 



^Nt = hNoe-(f-'M)t_ 



(2) 



Manuscript accepted 11 March 1992. 

 Fishery Bulletin, U.S. 90:412-416 (1992). 



412 



